Synchronization of Spin-Torque Oscillators via Continuation Method

In this paper we study synchronization phenomena for spin-torque oscillators coupled on a ring. Spin-torque oscillators are nanoelectronic devices which promise efficient microwave generation provided they are synchronized in large arrays. Due to their nonlinear and non-isochronous nature, their synchronization is difficult to analyse explicitly. We employ a recently developed continuation method that transforms the network of coupled oscillators (each described by an ordinary differential equation) into a single nonlinear partial differential equation (PDE). We then analyse the synchronization of this PDE in two cases: when all the oscillators are identical and when there are two different types of oscillators. In the case of identical oscillators we reconstruct all possible synchronous solutions in the system and provide explicit conditions for stability. For non-identical oscillators we derive and solve a differential synchronization condition which allows us to reconstruct the shape of the equilibrium profiles. All the presented results, which are derived for the PDE, are validated by numerical simulations of the original network of ODEs